The NASA Spaceflight article Japanese sounding rocket claims record-breaking orbital launch describes JAXA's use of its SS-520-4 sounding rocket to put the Tricom-1R cubesat (43201, 2018-016A) into orbit. While there was an issue related to communications during the ascent, the satellite is currently in a 1572 x 189 km orbit.
This is probably the smallest solid propellant rocket to put a payload into orbit. As pointed out by @PavelBernshtam answer this is certainly not the only example.
SS-520-4 is a three-stage solid-fueled rocket standing 9.54 meters tall, measuring 52 centimeters in diameter and weighing in at 2,600 Kilograms – smaller and lighter than any previous ground-based orbital launch vehicle. It is based on the SS-520 sounding rocket design, modified with a small third stage tasked with injecting a payload into Low Earth Orbit.
This is not the first attempt to reach orbit with the solid propellant SS-520-4 sounding rocket. See Will JAXA try again to launch TRICOM-1 with the “world's smallest orbital rocket” SS-520-4 again?
above left: Screenshot from N2YO.com showing Tricom-1R is indeed in orbit around the earth! Click for full size.
above right: SS-520-4 rocket ready for launch. From here, Photo: JAXA Click images for full size.
above: SS-520-4 rocket. From here, Image: JAXA